Question

Evaluate the expression when \(x=3, y=-2\), and \(z=4$$\frac{4 z-2 y}{20 x}\)

Short answer

The evaluated expression is \( \frac{1}{3} \).

Step-by-step solution

Substitution of the Given Values

Substitute \(x=3\), \(y=-2\), and \(z=4\) into the expression \(\frac{4 z-2 y}{20 x}\) to get \(\frac{4*4 - 2*(-2)}{20*3}\).

Simplification

Simplify the numerator and denominator separately to get \(\frac{16 + 4}{60}\).

Evaluate the Expression

Finally, perform the division operation to get the value \(\frac{20}{60}\).

Simplification

Simplify the fraction by dividing both numerator and denominator by their Greatest Common Divisor (GCD) which is 20, to get \( \frac{1}{3}\).

Key concepts

Algebraic Substitution

Algebraic substitution is a fundamental technique in algebra to evaluate expressions by replacing variables with their given numerical values. This method turns an algebraic expression into a simpler numerical one that can be computed. For instance, consider an expression \(\frac{4z - 2y}{20x}\), where each variable represents a certain value.

When given that \(x=3\), \(y=-2\), and \(z=4\), we substitute these numbers directly into the place of the corresponding variables. Thus, the expression becomes \(\frac{4*4 - 2*(-2)}{20*3}\). This first step is crucial as it sets up the expression for further simplification and final evaluation. \(x\), \(y\), and \(z\) are essentially placeholders for the numbers, and the substitution ensures that we are accurately reflecting the values they represent in our calculations.

Simplifying Fractions

Simplifying fractions is an essential skill to present the results of numerical expressions in their simplest form. After substituting values and performing operations simplification often involves finding equivalences that are less complex but have the same value. For the given expression, once we have substituted and performed the initial arithmetic, we get \(\frac{20}{60}\).

However, this fraction isn't in its simplest form. We want to express it in terms of the smallest possible whole numbers, both in the numerator and the denominator. To simplify, we find the Greatest Common Divisor (GCD) of both numbers. Since 20 divides evenly into 60, it becomes clear that the GCD is 20. We divide the numerator and the denominator by this GCD to obtain the simplified fraction \(\frac{1}{3}\). Simplification makes the results more understandable and is easier to work with in subsequent mathematical operations.

Numerical Expression Evaluation

Evaluating numerical expressions involves performing arithmetic operations like addition, subtraction, multiplication, and division in the correct order to arrive at a single numerical value. In the case of our expression \(\frac{20}{60}\), we've already substituted the variables with their values and simplified the terms.

The next step is to execute the division operation. Division might look simple, but it sometimes requires reducing fractions to their simplest form as we have done earlier. It's also vital to adhere to order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure accuracy. In simpler cases such as ours, once the expression is simplified, performing the division leads us to the final result, \(\frac{1}{3}\), which is the evaluated form of the numerical expression after carefully following each step.